Comment by voxl
7 hours ago
This is horseshit. Mathlib3 and mathlib4 all existed prior to LLMs. Unimath of Agda, mathematical components of Rocq, the list goes on.
LLMs have done nothing for "making large scale mechanization viable." They have been viable. The only thing has changed is the perception of the random developer who never wanted to put the effort into learning what actually needed to be learned and are instead happy to spit our complete garbage, spec and all, and say it's a proof of something.
It's not shocking at all that the people who seem to get any benefit out of LLMs in the proof assistant space are the ones who could have just don't it themselves anyway.
I actually noticed after I posted that I should have said 'by individuals at pace not too different from writing down a standard prose proof' or something similar to address this point. Mathlib for instance is of course a phenomenal project, but it was written by a large amount of people over many years. Granted their goal was not so much speed as it was elegance, but by 'viable' I had in mind something that a research mathematician could actually use in real time along actual math research. The content in Mathlib still falls very much short of being enough for formalizing much of the actual research being done in mathematics, but people are now autoformalizing non-trivial extensions on top of that, both individually and collaboratively (such as the recent Tau Ceti project).
Anyway, what do you mean by 'say it's a proof of something'? If you mean this in technical sense, there are plenty of examples around where single individuals or small teams have autoformalized theorems with tens or hundreds of thousands of lines Lean code that pass a comparator challenge (so the theorem is correct). If you mean this in the sense that the proof is also a proof in the eyes of humans, so that someone has actually read or understood the proof, I'm willing to acknowledge that in this area there is much work still to do. In my own experience the current LLMs are already very strong at formally proving theorems (with correct semantics), but they are still lacking in writing human-readable math prose based on these formalizations, for instance.
> The content in Mathlib still falls very much short of being enough for formalizing much of the actual research being done in mathematics, but people are now autoformalizing non-trivial extensions on top of that,
Formalising non-trivial extensions on top of the standard library is not hard. I could bash out an average textbook formalisation at about half reading speed. The hard part is making something elegant and general, which: and that's something that LLMs don't seem able to do.¹ (If you're lucky, the textbook you're working from has already distilled the best abstractions, and there's very little work left to do to make it good enough for a library: but such textbooks do not exist in research mathematics.)
> Granted their goal was not so much speed as it was elegance,
Do not underestimate the necessity of mathematical elegance. Mathematical notation is a tool powerful enough to teach machines to think: it is essential to make these tools elegant, or you will not be able to communicate your insight to your colleagues, and certainly not the next generation. Treating the goal of mathematics as "prove the most theorems as fast as possible" is hacking off the lower branches we should be using to climb trees, simply to harvest their fruit.
¹: Anyone familiar with my HN comment history will know that I keep banging on about "cannot in principle" and "there are deep theoretical reasons that an LLM can never". I'm not doing that here: I don't know a reason that LLMs can't produce elegant mathematical abstractions – probably because I don't understand mathematics deeply enough. "LLMs can't do this" is purely an empirical observation. (Believe me, I've read a lot of LLM-generated formal mathematics: it is invariably garbage. I just don't know why.)
(Sorry for the long post, but yours brought me thinking about a bunch of different aspects.)
First of all, I did not mean to downplay elegance at all. I agree that elegance is very important and that math is very much about trying to find elegant ways to think about various problems and phenomena. It also makes math feel more human and art-like, as elegance is not completely objective. And I also agree that LLMs do not seem to currently have consistent mathematical taste. I find they often do quite ugly or unoptimal proofs, although sometimes they also surprise me with a more elegant one than what I had in mind myself. And when we pass from arguments to choosing good definitions or seeing the big picture they are often much worse. Finally, formalization efforts such as Mathlib are very interesting from the elegance point of view. I'd actually be interested to see whether it would be possible to do lecture notes or textbooks based on Mathlib, written in standard math prose so that wider crowd of mathematicians might benefit from the insights that people had while formalizing.
However, as a research mathematician, I think we might now be approaching the situation where I can do my research pretty much as I usually do it, but at the same time in parallel have formalized proofs for the lemmas and theorems. These formalized versions are at least at the moment not going to have pretty proofs, and the proofs the LLM comes up with might even be different from what I'm writing in the paper (but probably in practice not very different if I'm formalizing every lemma). Still, if this can be done quickly enough, I think the result could be net positive even if the formalizations never leave my local hard drive: I will have confidence that I did not miss an edge case in the statements, where usually double-checking these things is actually a very time-consuming part of writing a paper. Thus I might be able to produce papers with less mistakes (usually non-important ones but they do happen). In this sort of workflow speed matters, and if you need a particular prerequisite theorem from the end of a textbook, you'd rather do it faster than half the reading speed (that's impressive by the way, and I do not mean this sarcastically!).
Returning a bit to the topic of elegance, I'd also like to claim that the elegance of arguments is probably at least as important as the elegance of definitions. And if you find an elegant argument, later on that might serve as a basis of a definition. There might also be some difference in how well this works out in practice in different fields. At least historically people in analysis (like myself) are happy to repeat known arguments in slightly different settings. It could be hard to make a version that works in every setting because different sets of assumptions could allow for a similar argument to work. Or it could also be easier to just remember the actual technique rather than trying to give it some jargonish name. Anyway, as I said above, I think LLMs are a bit better with arguments than definitions, so some elegance might be retained and perhaps you can later refactor to use more elegant definitions as well. (Hmm, a random idle thought, but a refactor from arguments to specific theorems could be in some sense similar as going from an untyped or not-explicitly-typed programming language to a typed one so there might be a coding analogue here as well.)
Finally, thanks for bringing up the limitations of LLMs. I also feel that LLMs are probably not currently able to really go beyond their training data, producing new theories with truly novel arguments or definitions. I'm skeptical that we will see a proof of the Riemann hypothesis in near future just drop from an LLM (human utilizing an LLM could be a bit of a different story, but I'm not a number theorist and have no idea whether anyone in the field has any plausible attack vectors currently). They are getting very good at combining and rephrasing existing stuff, however.
1 reply →
> LLMs have done nothing for "making large scale mechanization viable."
I'm sorry, but this assertion is ridiculous. LLMs absolutely have radically simplified mechanization. Autoformalization of papers using LLM is orders of magnitude faster than doing it manually.
To quote Ken Buzzard (of the Xena Project):
https://xenaproject.wordpress.com/2026/07/20/human-mathemati...
> [...] Sol had generated 1.2 million lines of Lean code in the three weeks that it had worked on the project. Lean’s fantastic (declaration of conflict of interest: I am a maintainer) mathematics library mathlib is only 2.3 million lines of code, and took nine years to write.
LLMs are going to be the basis of total formalization of all ~4M papers in the historical math literature. This project wouldn't be feasible without them.
What is absurd is evaluating something on the lines of code it produces. AI psychosis at work.
That's some industrial strength denial you've got going there. Please entertain the possibility that people who are presenting reasoned arguments with evidence against you are not "psychotic". BTW, is Ken Buzzard also psychotic?
Anyway, this avalanche is going to come down on your head whether or not you believe in it. Have fun!